Sun–Yang–Zuo conjecture on periodic Higgs bundles and torsion points

Let $X=cmathbb P^1$ and let $S$ be the branch locus of the double cover

$$\pi\colon C_\lambda\to X,$$

where $C_\lambda$ is the elliptic curve associated with the parameter $\lambda$. Let $\mathcal M_\lambda$ be the moduli space of the relevant logarithmic Higgs bundles, and let $\theta_0$ denote the zero of the Higgs field of a Higgs bundle in $\mathcal M_\lambda$. A Higgs bundle over a finite unramified extension is twisted $f$-periodic when its Higgs–de Rham flow returns to it after $f$ steps, up to the prescribed twist.

Sun–Yang–Zuo conjecture. A Higgs bundle in $\mathcal M_\lambda$ over a finite unramified extension is twisted $f$-periodic if and only if $\pi^*(\theta_0)$ is a $(p^f\pm1)$-torsion point in $C_\lambda$.

The conjecture links periodic Higgs–de Rham flows with torsion points on the associated elliptic curve. The source states that it has been checked modulo $p$ for $p\leq50$, for supersingular $C_{\bar\lambda}$ when $p\leq50$, and for torsion orders $1$, $2$, $3$, $4$, and $6$.